Why do Muslims only eat Halal meat and any other form of slaughtered meat is not permissible?

With respect to meat, the term Halal means "prepared appropriately for human consumption." The primary reason they do this is because they believe that God has instructed them to do so. There are other explanations from a scientific perspective (e.g. blood is probably unhealthy, pork is probably unhealthy), but even if Muslims didn't understand why it was this way, they'd do it out of obedience and trust in God.

Prohibited to you are dead animals, blood, the flesh of swine, and that which has been dedicated to other than Allah , and [those animals] killed by strangling or by a violent blow or by a head-long fall or by the goring of horns, and those from which a wild animal has eaten, except what you [are able to] slaughter [before its death], and those which are sacrificed on stone altars, and [prohibited is] that you seek decision through divining arrows. That is gr***e disobedience. This day those who disbelieve h***e despaired of [defeating] your religion; so fear them not, but fear Me. This day I h***e perfected for you your religion and completed My f***or upon you and h***e approved for you Islam as religion. But whoever is forced by severe hunger with no inclination to sin - then indeed, Allah is Forgiving and Merciful.

So, if a food meets these criteria, then it is considered acceptable to eat or Halal. In practice this means:

The animal was slaughtered by a Muslim, or by a Jew or a Christian (since they share the same God) and they must slaughter it in the name of God.

The slaughter involves cutting the carotid artery of the animal and letting the blood drain. This is not in this verse, but other sayings make this clear.

Death from a blow to the head, or fallen animals, are not permitted.

The other sources also indicate that one must be merciful to the animal (e.g. not letting it see other animals be slaughtered, and making sure the blade is sharp so the slaughter is painless.

Halal literally means permissible in Arabic and refers to foods that Allah has not forbidden in the Quran. Allah explicitly forbids pig's flesh, carrion (dead carcasses), blood, and animals sacrificed to a false god. All other meat is halal, as Allah reminds us in 5:87 "You who believe, do not forbid the good things God has made lawful to you."

Some Muslims confuse the guidance on carrion with a commandment that all meat be slaughtered in a specific way. Allah makes it very clear that he wants Muslims to only consume meat that was killed with the intention to be eaten. That is why Allah says in 5:1

...Livestock animals are lawful as food for you, with the exception of what is about to be announced to you...

The livestock requirement is important, because livestock is raised and killed to be eaten. Allah then explains in 5:3

You are forbidden to eat carrion; blood; pig's meat; any animal over which any name other than God's has been invoked; any animal strangled, or victim of a violent blow or fall, or gored or s***aged by a beast of prey, unless you still slaughter it

So animals that were accidentally maimed-injured (ie potentially carrion) could only be consumed if the animal could still be killed with the intention to be eaten. Allah was very particular about this point and repeated it many times at 2:173, 5:3, 6:145, and 16:115

These rules opened Muslims to ridicule from non-Muslims during the Prophet's lifetime as explained in Wahidi's Asbab al-Nuzul (1075 CE):

Said 'Ikrimah: “When Allah, exalted is He, revealed the unlawfulness of the meat of carrion, the Magians ...wrote...that Muhammad and his Companions claim that they follow the command of Allah but then say that what they slaughter is lawful and that which is slaughtered by Allah is unlawful. This caused doubt in the minds of some Muslims...The idolaters said: “O Muhammad, tell us: 'When a sheep dies, who has killed it?' He said: 'Allah killed it!' They said: 'How is it then that that which is killed by you and your Companions is lawful, that which is killed by a dog or bird of prey is lawful but that which is killed by Allah is unlawful?'

The Magians were essentially questioning why Muslims refused to eat carrion, since carrion was theoretically killed by Allah himself. These questions caused Muslims to wonder if carrion was allowed to be eaten after all. Allah replied to this question in 6:118-122, re-affirming that carrion can only be eaten if you h***e enough time to slaughter the animal and pronounce Allah's name over it:

So [believers] eat any [animal] over which God’s name has been pronounced, if you believe in His revelations. Why should you not eat such animals when God has already fully explained what He has forbidden you, except when forced by hunger? But many lead others astray by their desires, without any true knowledge: your Lord knows best who oversteps the limit. ***oid committing sin, whether openly or in secret, for those who commit sin will be repaid for what they do, and do not eat anything over which God’s name has not been pronounced, for that is breaking the law. The evil ones incite their followers to argue with you: if you listen to them, you too will become idolaters.

The verse is basically saying 'if you encounter a dying animal, slaughter it in the name of Allah, and don't eat it if the animal is already dead. Idolators are trying to trick you into eating carrion, don't do it, because that is breaking God's law!'

Although this verse is clearly referring only to carrion situations, some scholars attempt to pull it out of context to require a ritual slaughter of all animals.The verse explicitly rejects this attempt by saying 'God has already fully explained what He has forbidden you.' These scholars h***e invented intricate rules of dhabiha (slaughter), specifying the sharpness of the blade used, the direction the animal is facing, a specific prayer to be recited, and the faith of the slaughterer. None of these rules appear in the Quran but are built on top of hadith traditions. Halal meat shops tend to be very focused on these rules as they provide a reason to charge a premium for their meats to Muslims who believe that they can only consume animals killed in this fashion.

Another very popular interpretation of this verse is that a Muslim merely needs to recite God's name before consuming (not slaughtering) any kind of permissible meat to satisfy it's requirement.